Chapter 3.1, Problem 40E

(a)

To determine

To Find: the mean number for states east of Mississippi river.

Answer to Problem 40E

Mean number of breweries for states east of Mississippi river $\approx 29.63$

Explanation of Solution

Given:

  $18,10,47,22,52,13,38,23,40,16,20,76,46,14,19,20$

Formula Used:

  $\begin{array}{l}\text{Mean}(\overline{x})=\frac{{\displaystyle \sum x_i}}{n}\\ \\ \text{Median}=\left\{\begin{array}{l}\frac{{\left(n+1\right)}^{\text{th }}\text{term}}{2},\text{When}\text{n}\text{is}\text{odd}\\ \frac{{\left(\frac{n}{2}+1\right)}^{\text{th }}\text{term + }{\left(\frac{n}{2}\right)}^{\text{th}}\text{term}}{2},\text{When n is even}\end{array}\right\}\end{array}$

Calculation:

Mean number of breweries for states east of Mississippi river

  $\begin{array}{c}\text{Mean}\left(\overline{x}\right)=\frac{{\displaystyle \sum x_i}}{n}\\ =\frac{\left[\begin{array}{l}18+10+47+22+52+13+38+23+\\ 40+16+20+76+46+14+19+20\end{array}\right]}{16}\\ =\frac{474}{16}\\ =29.625\approx 29.63\end{array}$

Therefore, for states east of Mississippi river, mean number of breweries 29.63.

(b)

To determine

To Find: the mean number for states west of Mississippi river.

Answer to Problem 40E

Mean number of breweries for states west of Mississippi river $\approx 56.27$

Explanation of Solution

Calculation- Mean number of breweries for states west of Mississippi river Mean, $\overline{x}$ ,

  $\begin{array}{c}\overline{x}=\frac{\sum x_i}{n}\\ =\frac{17+31+305+111+21+6+41+30+5+37+15}{11}\\ =\frac{619}{11}=56.2727\approx 56.27\end{array}$

Therefore, for states west of Mississippi river, mean number of breweries $\approx 56.27$ .

(c)

To determine

To Find: the median number for states east of Mississippi river.

Answer to Problem 40E

Median number of breweries for states east of Mississippi river $=21$

Explanation of Solution

Calculation-

Median number of breweries for states east of Mississippi river

For median, it is required to first arrange the data in sequential order,

We have,

  $10,13,14,16,18,19,20,20,22,23,38,40,46,47,52,76$

Median, for even numbers,

  $\begin{array}{l}\text{=}\left(\frac{{\left(\frac{\text{16}}{\text{2}}\right)}^{\text{th}}\text{term+}{\left(\frac{\text{16}}{\text{2}}\text{+1}\right)}^{\text{th}}\text{term}}{\text{2}}\right)\\ \text{=}\left(\frac{{\left(\text{8}\right)}^{\text{th}}\text{term+}{\left(\text{9}\right)}^{\text{th}}\text{term}}{\text{2}}\right)\\ \text{=}\left(\frac{\text{20+22}}{\text{2}}\right)\\ \text{=}\frac{\text{42}}{\text{2}}\\ \text{=21}\end{array}$

Therefore, for states east of Mississippi river, median number of breweries $=21$

(d)

To determine

To Find: the median number for states west of Mississippi river.

Answer to Problem 40E

Median number of breweries for states west of Mississippi river $=30$

Explanation of Solution

Calculation-

Median number of breweries for states west of Mississippi river

For median, first arrange the data in sequential order,

We have,

  $10,13,14,16,18,19,20,20,22,23,38,40,46,47,52,76$

Median, for odd numbers,

  $\begin{array}{l}\text{=}{\left(\frac{\left(\text{n+1}\right)}{\text{2}}\right)}^{\text{th}}\text{term}\\ \text{=}{\left(\frac{\left(\text{11+1}\right)}{\text{2}}\right)}^{\text{th}}\text{term}\\ \text{=}{\left(\frac{\left(\text{12}\right)}{\text{2}}\right)}^{\text{th}}\text{term}\\ {\text{=6}}^{\text{th}}\text{term}\\ \text{=30}\end{array}$

Therefore, for states west of Mississippi river, median number of breweries $=30$

(e)

To determine

To Find: Whether one region have a lot more breweries per state than the other.

Answer to Problem 40E

Yes, west region has a lot more breweries per state than east region.

Explanation of Solution

By looking at the mean number of breweries for states east and west of Mississippi river,

Mean number of breweries for states east of Mississippi river $\approx 29.63$

Mean number of breweries for states west of Mississippi river $\approx 56.27$

So, west region has a lot more breweries per state than east region.

(f)

To determine

To Find: the mean and median number for states west of Mississippi river without including California.

Answer to Problem 40E

Mean number of breweries for rest of states west of Mississippi river $=31.4$

Median number of breweries for rest of states west of Mississippi river $=25.5$

Explanation of Solution

Calculation-

If California is removed from sample of western states, then to find mean and median, the sample of data of western states is to be rearranged.

The new sample of data is,

  $17,31,111,21,6,41,30,5,37,15$

Mean

  $\begin{array}{l}=\frac{17+31+111+21+6+41+30+5+37+15}{10}\\ =\frac{314}{10}=31.4\end{array}$

Median

  $\begin{array}{c}\text{Median =}\left(\frac{{\left(\frac{\text{10}}{\text{2}}\right)}^{\text{th}}\text{term+}{\left(\frac{\text{10}}{\text{2}}\text{+1}\right)}^{\text{th}}\text{term}}{\text{2}}\right)\\ \text{=}\left(\frac{{\left(\text{5}\right)}^{\text{th}}\text{term+}{\left(\text{6}\right)}^{\text{th}}\text{term}}{\text{2}}\right)\\ \text{=}\left(\frac{\text{21+30}}{\text{2}}\right)\\ \text{=}\frac{\text{51}}{\text{2}}\\ \text{=25}\text{.5}\end{array}$

Therefore, for rest of states west of Mississippi river, mean number of breweries $=31.4$ and median number of breweries $=25.5$ .

(g)

To determine

To Find: the number for breweries between eastern and western states without including California.

Answer to Problem 40E

West region has a greater number of breweries per state than east region.

Explanation of Solution

By looking at the mean number of breweries per state of east region and mean number of breweries per state of west region without including California,

Mean number of breweries for states east of Mississippi river $\approx 29.63$

Mean number of breweries for states west of Mississippi river without including California $=31.4$

So, it suggests that even if California is removed from western region, then also west region has a greater number of breweries per state than east region.